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Journal articles on the topic 'Commuting graph'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Nath, Rajat Kanti, and Jutirekha Dutta. "Spectrum of commuting graphs of some classes of finite groups." MATEMATIKA 33, no. 1 (2017): 87. http://dx.doi.org/10.11113/matematika.v33.n1.812.

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In this paper, we initiate the study of spectrum of the commuting graphs of finite non-abelian groups. We first compute the spectrum of this graph for several classes of finite groups, in particular AC-groups. We show that the commuting graphs of finite non-abelian AC-groups are integral. We also show that the commuting graph of a finite non-abelian group G is integral if G is not isomorphic to the symmetric group of degree 4 and the commuting graph of G is planar. Further, it is shown that the commuting graph of G is integral if its commuting graph is toroidal.

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2

GIUDICI, MICHAEL, and BOJAN KUZMA. "REALIZABILITY PROBLEM FOR COMMUTING GRAPHS." Journal of the Australian Mathematical Society 101, no. 3 (2016): 335–55. http://dx.doi.org/10.1017/s1446788716000148.

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We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain complete classifications of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.

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3

Alimon, Nur Idayu, Nor Haniza Sarmin, and Ahmad Erfanian. "Topological indices of non-commuting graph of dihedral groups." Malaysian Journal of Fundamental and Applied Sciences 14 (October 25, 2018): 473–76. http://dx.doi.org/10.11113/mjfas.v14n0.1270.

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Assume is a non-abelian group A dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. The non-commuting graph of denoted by is the graph of vertex set whose vertices are non-central elements, in which is the center of and two distinct vertices and are joined by an edge if and only if In this paper, some topological indices of the non-commuting graph, of the dihedral groups, are presented. In order to determine the Edge-Wiener index, First Zagreb index and Second Zagreb index of the non-commuting graph, of the dihedral groups, previous results of some of the topological indices of non-commuting graph of finite group are used. Then, the non-commuting graphs of dihedral groups of different orders are found. Finally, the generalisation of Edge-Wiener index, First Zagreb index and Second Zagreb index of the non-commuting graphs of dihedral groups are determined.

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4

Leshchenko, Yu Yu, and L. V. Zoria. "On diameters estimations of the commuting graphs of Sylow $p$-subgroups of the symmetric groups." Carpathian Mathematical Publications 5, no. 1 (2013): 70–78. http://dx.doi.org/10.15330/cmp.5.1.70-78.

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The commuting graph of a group $G$ is an undirected graph whose vertices are non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if and only if $xy=yx$. This article deals with the properties of the commuting graphs of Sylow $p$-subgroups of the symmetric groups. We define conditions of connectedness of respective graphs and give estimations of the diameters if graph is connected.

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5

Dutta, Jutirekha, Walaa Nabil Taha Fasfous, and Rajat Kanti Nath. "Spectrum and genus of commuting graphs of some classes of finite rings." Acta et Commentationes Universitatis Tartuensis de Mathematica 23, no. 1 (2019): 5–12. http://dx.doi.org/10.12697/acutm.2019.23.01.

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We consider commuting graphs of some classes of finite rings and compute their spectrum and genus. We show that the commuting graph of a finite CC-ring is integral. We also characterize some finite rings whose commuting graphs are planar.

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6

Ahmadidelir, Karim. "On the non-commuting graph in finite Moufang loops." Journal of Algebra and Its Applications 17, no. 04 (2018): 1850070. http://dx.doi.org/10.1142/s0219498818500706.

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The non-commuting graph associated to a non-abelian group [Formula: see text], [Formula: see text], is a graph with vertex set [Formula: see text] where distinct non-central elements [Formula: see text] and [Formula: see text] of [Formula: see text] are joined by an edge if and only if [Formula: see text]. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text]. Then [Formula: see text]. Conjecture 2 (AAM’s Conjecture). Let [Formula: see text] be a finite non-abelian simple group and [Formula: see text] be a group such that [Formula: see text]. Then [Formula: see text]. Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop [Formula: see text] and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

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7

OMIDI, G. R., and E. VATANDOOST. "ON THE COMMUTING GRAPH OF RINGS." Journal of Algebra and Its Applications 10, no. 03 (2011): 521–27. http://dx.doi.org/10.1142/s0219498811004811.

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Let R be a non-commutative ring. The commuting graph of R denoted by Γ(R), is a graph with vertex set R\Z(R) and two vertices a and b are adjacent if ab = ba. It has been shown that the diameter of Γ(R)c is less than 3. For a finite ring R we show that the diameter of Γ(R)c is one if and only if R is the non-commutative ring on 4 elements. Also we characterize all rings where the complements of their commuting graphs are planar. Moreover, we identify the commuting graphs of rings of order pi for i = 2, 3 and prime number p.

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8

Ghorbani, Modjtaba, and Zahra Gharavi-Alkhansari. "A note on integral non-commuting graphs." Filomat 31, no. 3 (2017): 663–69. http://dx.doi.org/10.2298/fil1703663g.

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The non-commuting graph ?(G) of group G is a graph with the vertex set G - Z(G) and two distinct vertices x and y are adjacent whenever xy ? yx. The aim of this paper is to study integral regular non-commuting graphs of valency at most 16.

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9

Bello, Muhammed, Nor Muhainiah Mohd Ali, and Surajo Ibrahim Isah. "Graph coloring using commuting order product prime graph." Journal of Mathematics and Computer Science 23, no. 02 (2020): 155–69. http://dx.doi.org/10.22436/jmcs.023.02.08.

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The concept of graph coloring has become a very active field of research that enhances many practical applications and theoretical challenges. Various methods have been applied in carrying out this study. Let G be a finite group. In this paper, we introduce a new graph of groups, which is a commuting order product prime graph of finite groups as a graph having the elements of G as its vertices and two vertices are adjacent if and only if they commute and the product of their order is a prime power. This is an extension of the study for order product prime graph of finite groups. The graph's general presentations on dihedral groups, generalized quaternion groups, quasi-dihedral groups, and cyclic groups have been obtained in this paper. Moreover, the commuting order product prime graph on these groups has been classified as connected, complete, regular, or planar. These results are used in studying various and recently introduced chromatic numbers of graphs.

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10

Rahayuningtyas, Handrini, Abdussakir Abdussakir, and Achmad Nashichuddin. "Bilangan Kromatik Grap Commuting dan Non Commuting Grup Dihedral." CAUCHY 4, no. 1 (2015): 16. http://dx.doi.org/10.18860/ca.v4i1.3169.

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Commuting graph is a graph that has a set of points X and two different vertices to be connected directly if each commutative in G. Let G non abelian group and Z(G) is a center of G. Noncommuting graph is a graph which the the vertex is a set of G\Z(G) and two vertices x and y are adjacent if and only if xy≠yx. The vertex colouring of G is giving k colour at the vertex, two vertices that are adjacent not given the same colour. Edge colouring of G is two edges that have common vertex are coloured with different colour. The smallest number k so that a graph can be coloured by assigning k colours to the vertex and edge called chromatic number. In this article, it is available the general formula of chromatic number of commuting and noncommuting graph of dihedral group

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11

Abdussakir, Abdussakir, Rivatul Ridho Elvierayani, and Muflihatun Nafisah. "On the Spectra of Commuting and Non Commuting Graph on Dihedral Group." CAUCHY 4, no. 4 (2017): 176. http://dx.doi.org/10.18860/ca.v4i4.4211.

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Study about spectra of graph has became interesting work as well as study about commuting and non commuting graph of a group or a ring. But the study about spectra of commuting and non commuting graph of dihedral group has not been done yet. In this paper, we investigate adjacency spectrum, Laplacian spectrum, signless Laplacian spectrum, and detour spectrum of commuting and non commuting graph of dihedral group <em>D</em><sub>2<em>n</em></sub>

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12

Hazlewood, Robert, Iain Raeburn, Aidan Sims, and Samuel B. G. Webster. "Remarks on some fundamental results about higher-rank graphs and their C*-algebras." Proceedings of the Edinburgh Mathematical Society 56, no. 2 (2013): 575–97. http://dx.doi.org/10.1017/s0013091512000338.

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AbstractResults of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

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13

Chen, Jing, and Lang Tang. "The Commuting Graphs on Dicyclic Groups." Algebra Colloquium 27, no. 04 (2020): 799–806. http://dx.doi.org/10.1142/s1005386720000668.

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For a group G and a non-empty subset Ω of G, the commuting graph [Formula: see text] of Ω is a graph whose vertex set is Ω and any two vertices are adjacent if and only if they commute in G. Define [Formula: see text], the dicyclic group of order [Formula: see text] [Formula: see text], which is also known as the generalized quaternion group. We mainly investigate the properties and metric dimension of the commuting graphs on the dicyclic group [Formula: see text].

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14

Nath, Rajat Kanti, Walaa Nabil Taha Fasfous, Kinkar Chandra Das, and Yilun Shang. "Common Neighborhood Energy of Commuting Graphs of Finite Groups." Symmetry 13, no. 9 (2021): 1651. http://dx.doi.org/10.3390/sym13091651.

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The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and only if xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) and the common neighborhood energy Ecn(G) of a simple graph G. A graph G is called CN-hyperenergetic if Ecn(G)>Ecn(Kn), where n=|V(G)| and Kn denotes the complete graph on n vertices. Two graphs G and H with equal number of vertices are called CN-equienergetic if Ecn(G)=Ecn(H). In this paper we compute the common neighborhood energy of Γc(G) for several classes of finite non-abelian groups, including the class of groups such that the central quotient is isomorphic to group of symmetries of a regular polygon, and conclude that these graphs are not CN-hyperenergetic. We shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic.

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15

HEGARTY, PETER, and DMITRII ZHELEZOV. "On the Diameters of Commuting Graphs Arising from Random Skew-Symmetric Matrices." Combinatorics, Probability and Computing 23, no. 3 (2014): 449–59. http://dx.doi.org/10.1017/s0963548313000655.

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We present a two-parameter family $(G_{m,k})_{m, k \in \mathbb{N}_{\geq 2}}$, of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős–Rényi random graph. As well as being of independent interest, our groups would, if our conjecture is true, provide a large family of counterexamples to the conjecture of Iranmanesh and Jafarzadeh that the commuting graph of a finite group, if connected, must have a bounded diameter. Simulations of our model yielded explicit examples of groups whose commuting graphs have all diameters from 2 up to 10.

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16

Bhat, K. Arathi, and G. Sudhakara. "Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )." Special Matrices 6, no. 1 (2018): 343–56. http://dx.doi.org/10.1515/spma-2018-0028.

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Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

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17

Abdussakir, Abdussakir. "Radius, Diameter, Multiplisitas Sikel, dan Dimensi Metrik Graf Komuting dari Grup Dihedral." Jurnal Matematika "MANTIK" 3, no. 1 (2017): 1–4. http://dx.doi.org/10.15642/mantik.2017.3.1.1-4.

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Commuting graph C(G) of a non-Abelian group G is a graph that contains all elements of G as its vertex set and two distinct vertices in C(G) will be adjacent if they are commute in G. In this paper we discuss commuting graph of dihedral group D2n. We show radius, diameter, cycle multiplicity, and metric dimension of this commuting graph in several theorems with their proof.

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18

IRANMANESH, A., and A. JAFARZADEH. "ON THE COMMUTING GRAPH ASSOCIATED WITH THE SYMMETRIC AND ALTERNATING GROUPS." Journal of Algebra and Its Applications 07, no. 01 (2008): 129–46. http://dx.doi.org/10.1142/s0219498808002710.

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The commuting graph of a group G, denoted by Γ(G), is a simple undirected graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = yx. The commuting graph of a subset of a group is defined similarly. In this paper we investigate the properties of the commuting graph of the symmetric and alternating and subsets of transpositions and involutions in the symmetric groups.

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19

Erfanian, A., K. Khashyarmanesh, and Kh Nafar. "Non-commuting graphs of rings." Discrete Mathematics, Algorithms and Applications 07, no. 03 (2015): 1550027. http://dx.doi.org/10.1142/s1793830915500275.

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Let R be a non-commutative ring and let C(R) be the center of R. The non-commuting graph ΓR of R associated to R is defined as a graph with R\C(R) as the vertices and two distinct vertices x and y are joined whenever xy ≠ yx. We study various graph theoretical properties of this graph.

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20

Hayat, Umar, Mubasher Umer, Ivan Gutman, Bijan Davvaz, and Álvaro Nolla de Celis. "A novel method to construct NSSD molecular graphs." Open Mathematics 17, no. 1 (2019): 1526–37. http://dx.doi.org/10.1515/math-2019-0129.

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Abstract A graph is said to be NSSD (=non-singular with a singular deck) if it has no eigenvalue equal to zero, whereas all its vertex-deleted subgraphs have eigenvalues equal to zero. NSSD graphs are of importance in the theory of conductance of organic compounds. In this paper, a novel method is described for constructing NSSD molecular graphs from the commuting graphs of the Hv-group. An algorithm is presented to construct the NSSD graphs from these commuting graphs.

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21

Wang, Lingli, and Wujie Shi. "Characterization of Aut(J2) and Aut(McL) by Their Non-commuting Graphs." Algebra Colloquium 18, no. 02 (2011): 327–32. http://dx.doi.org/10.1142/s1005386711000228.

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For a non-abelian group G, we associate the non-commuting graph ∇ (G) whose vertex set is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this paper, we prove that Aut (J2) and Aut (McL) are characterized by their non-commuting graphs.

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22

AKBARI, M., and A. R. MOGHADDAMFAR. "THE EXISTENCE OR NONEXISTENCE OF NON-COMMUTING GRAPHS WITH PARTICULAR PROPERTIES." Journal of Algebra and Its Applications 13, no. 01 (2013): 1350064. http://dx.doi.org/10.1142/s0219498813500643.

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We consider the non-commuting graph ∇(G) of a non-abelian finite group G; its vertex set is G\Z(G), the set of non-central elements of G, and two distinct vertices x and y are joined by an edge if [x, y] ≠ 1. We determine the structure of any finite non-abelian group G (up to isomorphism) for which ∇(G) is a complete multipartite graph (see Propositions 3 and 4). It is also shown that a non-commuting graph is a strongly regular graph if and only if it is a complete multipartite graph. Finally, it is proved that there is no non-abelian group whose non-commuting graph is self-complementary and n-cube.

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23

Dutta, Jutirekha, Dhiren K. Basnet, and Rajat K. Nath. "On Generalized Non-commuting Graph of a Finite Ring." Algebra Colloquium 25, no. 01 (2018): 149–60. http://dx.doi.org/10.1142/s100538671800010x.

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Let S and K be two subrings of a finite ring R. Then the generalized non-commuting graph of subrings S, K of R, denoted by ГS,K, is a simple graph whose vertex set is [Formula: see text], and where two distinct vertices a, b are adjacent if and only if [Formula: see text] or [Formula: see text] and [Formula: see text]. We determine the diameter, girth and some dominating sets for ГS,K. Some connections between ГS,K and Pr(S, K) are also obtained. Further, ℤ-isoclinism between two pairs of finite rings is defined, and we show that the generalized non-commuting graphs of two ℤ-isoclinic pairs are isomorphic under some conditions.

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24

TOLUE, B., and A. ERFANIAN. "RELATIVE NON-COMMUTING GRAPH OF A FINITE GROUP." Journal of Algebra and Its Applications 12, no. 02 (2012): 1250157. http://dx.doi.org/10.1142/s0219498812501575.

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The essence of the non-commuting graph remind us to find a connection between this graph and the commutativity degree as denoted by d(G). On the other hand, d(H, G) the relative commutativity degree, was the key to generalize the non-commuting graph ΓG to the relative non-commuting graph (denoted by ΓH, G) for a non-abelian group G and a subgroup H of G. In this paper, we give some results about ΓH, G which are mostly new. Furthermore, we prove that if (H1, G1) and (H2, G2) are relative isoclinic then ΓH1, G1 ≅ Γ H2, G2 under special conditions.

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Afkhami, M., Z. Barati, N. Hoseini, and K. Khashyarmanesh. "A generalization of commuting graphs." Discrete Mathematics, Algorithms and Applications 07, no. 01 (2015): 1450068. http://dx.doi.org/10.1142/s1793830914500682.

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Let R be a ring with the identity element 1, α be an endomorphism of R and δ be a left α-derivation. In this paper, we introduce a generalization of a commuting graph, which is denoted by ΓR(α, δ), as a directed graph with vertex set R and, for two distinct vertices x and y, there is an arc from x to y if and only if xy = α(y)x + δ(y). We study some basic properties of ΓR(α, δ). Also, we investigate the planarity and genus of the graph ΓR(α, 0).

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26

Abdollahi, A., S. Akbari, and H. R. Maimani. "Non-commuting graph of a group." Journal of Algebra 298, no. 2 (2006): 468–92. http://dx.doi.org/10.1016/j.jalgebra.2006.02.015.

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Jahandideh, M., R. Modabernia, and S. Shokrolahi. "Non-commuting graphs of certain almost simple groups." Asian-European Journal of Mathematics 12, no. 05 (2019): 1950081. http://dx.doi.org/10.1142/s1793557119500815.

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Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

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Kakkar, Vipul, and Gopal Singh Rawat. "Commuting graphs of generalized dihedral groups." Discrete Mathematics, Algorithms and Applications 11, no. 02 (2019): 1950024. http://dx.doi.org/10.1142/s1793830919500241.

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For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text], the commuting graph of [Formula: see text], denoted by [Formula: see text] is the graph whose vertex set is [Formula: see text] and any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if they commute in [Formula: see text]. In this paper, certain properties of the commuting graph of generalized dihedral groups have been studied.

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29

Planat, M., and M. Saniga. "On the Pauli graphs on N-qudits." Quantum Information and Computation 8, no. 1&2 (2008): 127–46. http://dx.doi.org/10.26421/qic8.1-2-9.

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A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of $N$-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding {\it Pauli graph} are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) $N$-qubit Pauli graph is shown to be pseudo-geometric, i.\,e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle $Q(4,3)$, the dual of $W(3)$.

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Devhare, Sarika, and Vinayak Joshi. "Perfect non-commuting graphs of matrices over chains." Discrete Mathematics, Algorithms and Applications 09, no. 04 (2017): 1750049. http://dx.doi.org/10.1142/s1793830917500495.

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In this paper, we study the non-commuting graph [Formula: see text] of strictly upper triangular [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We prove that [Formula: see text] is a compact graph. From [Formula: see text], we construct a poset [Formula: see text]. We further prove that [Formula: see text] is a dismantlable lattice and its zero-divisor graph is isomorphic to [Formula: see text]. Lastly, we prove that [Formula: see text] is a perfect graph.

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Dolzan, David, Damjana Kokol-Bukovsek, and Bojan Kuzma. "On the lower bound for diameter of commuting graph of prime-square sized matrices." Filomat 32, no. 17 (2018): 5993–6000. http://dx.doi.org/10.2298/fil1817993d.

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It is known that the diameter of commuting graph of n-by-n matrices is bounded above by six if the graph is connected. In the commuting graph of p2-by-p2 matrices over a sufficiently large field which admits a cyclic Galois extension of degree p2 we construct two matrices at distance at least five. This shows that five is the lower bound for its diameter. Our results are applicable for all sufficiently large finite fields as well as for the field of rational numbers.

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32

Parker, Christopher. "The commuting graph of a soluble group." Bulletin of the London Mathematical Society 45, no. 4 (2013): 839–48. http://dx.doi.org/10.1112/blms/bdt005.

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Ali, Faisal, Muhammad Salman, and Shuliang Huang. "On the Commuting Graph of Dihedral Group." Communications in Algebra 44, no. 6 (2016): 2389–401. http://dx.doi.org/10.1080/00927872.2015.1053488.

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34

Vatandoost, Ebrahim, and Yasser Golkhandypour. "Domination in Commuting Graph and its Complement." Iranian Journal of Science and Technology, Transactions A: Science 41, no. 2 (2016): 383–91. http://dx.doi.org/10.1007/s40995-016-0028-5.

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35

Darafsheh, M. R. "Groups with the same non-commuting graph." Discrete Applied Mathematics 157, no. 4 (2009): 833–37. http://dx.doi.org/10.1016/j.dam.2008.06.010.

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36

Kumar, Jitender, Sandeep Dalal, and Vedant Baghel. "On the Commuting Graph of Semidihedral Group." Bulletin of the Malaysian Mathematical Sciences Society 44, no. 5 (2021): 3319–44. http://dx.doi.org/10.1007/s40840-021-01111-0.

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37

Suleiman, Aliyu, and Aliyu Ibrahim Kiri. "The independence polynomial of inverse commuting graph of dihedral groups." Malaysian Journal of Fundamental and Applied Sciences 16, no. 1 (2020): 115–20. http://dx.doi.org/10.11113/mjfas.v16n1.1353.

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Set of vertices not joined by an edge in a graph is called the independent set of the graph. The independence polynomial of a graph is a polynomial whose coefficient is the number of independent sets in the graph. In this research, we introduce and investigate the inverse commuting graph of dihedral groups (D2N) denoted by GIC. It is a graph whose vertex set consists of the non-central elements of the group and for distinct x,y, E D2N, x and y are adjacent if and only if xy = yx = 1 where 1 is the identity element. The independence polynomials of the inverse commuting graph for dihedral groups are also computed. A formula for obtaining such polynomials without getting the independent sets is also found, which was used to compute for dihedral groups of order 18 up to 32.

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38

Iranmanesh, Mahdiyeh, Morteza Jafarpour, and Irina Cristea. "The non-commuting graph of a non-central hypergroup." Open Mathematics 17, no. 1 (2019): 1035–44. http://dx.doi.org/10.1515/math-2019-0084.

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Abstract The aim of this paper is to construct and study the properties of a certain graph associated with a non-central hypergroup, i.e. a hypergroup having non-commutative the associated fundamental group. The method of the construction of the graph is similar to that one proposed by Paul Erdős, when he defined a graph associated with a non-commutative group. We establish necessary and /or sufficient conditions for the associated graph to be Hamiltonian or planar.

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39

Abdollahi, A., A. Azad, A. Mohammadi Hassanabadi, and M. Zarrin. "On the Clique Numbers of Non-commuting Graphs of Certain Groups." Algebra Colloquium 17, no. 04 (2010): 611–20. http://dx.doi.org/10.1142/s1005386710000581.

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Let G be a non-abelian group. The non-commuting graph [Formula: see text] of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with [Formula: see text], where 57 is the clique number of the non-commuting graph of the projective special linear group PSL (2,7). We also determine [Formula: see text] for all finite minimal simple groups G.

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40

Zhang, Hengbin. "Automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $." AIMS Mathematics 6, no. 11 (2021): 12650–59. http://dx.doi.org/10.3934/math.2021729.

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<abstract><p>Let $ R $ be a ring with identity. The commuting graph of $ R $ is the graph associated to $ R $ whose vertices are non-central elements in $ R $, and distinct vertices $ A $ and $ B $ are adjacent if and only if $ AB = BA $. In this paper, we completely determine the automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $, where $ \mathbb{Z}_{p^{s}} $ is the ring of integers modulo $ p^{s} $, $ p $ is a prime and $ s $ is a positive integer.</p></abstract>

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41

PAKOVICH, FEDOR. "Commuting rational functions revisited." Ergodic Theory and Dynamical Systems 41, no. 1 (2019): 295–320. http://dx.doi.org/10.1017/etds.2019.51.

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Let $B$ be a rational function of degree at least two that is neither a Lattès map nor conjugate to $z^{\pm n}$ or $\pm T_{n}$. We provide a method for describing the set $C_{B}$ consisting of all rational functions commuting with $B$. Specifically, we define an equivalence relation $\underset{B}{{\sim}}$ on $C_{B}$ such that the quotient $C_{B}/\underset{B}{{\sim}}$ possesses the structure of a finite group $G_{B}$, and describe generators of $G_{B}$ in terms of the fundamental group of a special graph associated with $B$.

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Khasraw, Sanhan Muhammad Salih, Ivan Dler Ali, and Rashad Rashid Haji. "On the non-commuting graph of dihedral group." Electronic Journal of Graph Theory and Applications 8, no. 2 (2020): 233. http://dx.doi.org/10.5614/ejgta.2020.8.2.3.

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Mirzargar, M., P. P. Pach, and Ali Reza Ashrafi. "Remarks On Commuting Graph of a Finite Group." Electronic Notes in Discrete Mathematics 45 (January 2014): 103–6. http://dx.doi.org/10.1016/j.endm.2013.11.020.

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Woodcock, Timothy. "The commuting graph of the symmetric group Sn." International Journal of Contemporary Mathematical Sciences 10 (2015): 287–309. http://dx.doi.org/10.12988/ijcms.2015.4553.

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Araújo, João, Wolfram Bentz, and Konieczny Janusz. "The commuting graph of the symmetric inverse semigroup." Israel Journal of Mathematics 207, no. 1 (2015): 103–49. http://dx.doi.org/10.1007/s11856-015-1173-9.

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Tolue, Behnaz. "The twin non-commuting graph of a group." Rendiconti del Circolo Matematico di Palermo Series 2 69, no. 2 (2019): 591–99. http://dx.doi.org/10.1007/s12215-019-00421-4.

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Tolue, Behnaz. "On the non-commuting graph of a group." MATHEMATICA 61 (84), no. 2 (2019): 190–97. http://dx.doi.org/10.24193/mathcluj.2019.2.09.

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Abdussakir, Dini Chandra Aulia Putri, and Ziyadatur Rohmah Fadhillah. "Full automorphism group of commuting and non-commuting graph of dihedral and symmetric groups." Journal of Physics: Conference Series 1028 (June 2018): 012112. http://dx.doi.org/10.1088/1742-6596/1028/1/012112.

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PATTABIRAMAN, K. "Distance based topological descriptors for two classes of graphs." Creative Mathematics and Informatics 28, no. 2 (2019): 151–62. http://dx.doi.org/10.37193/cmi.2019.02.07.

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In this paper, the exact formula for the generalized product degree distance, reciprocal product degree distance and product degree distance of Mycielskian graph and its complement are obtained. In addition, we compute the above indices for non-commuting graph.

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50

GENG, XIANYA, LITING FAN, and XIAOBIN MA. "DIAMETER OF COMMUTING GRAPHS OF SYMPLECTIC ALGEBRAS." Bulletin of the Australian Mathematical Society 100, no. 3 (2019): 419–27. http://dx.doi.org/10.1017/s0004972719000583.

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Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$, where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$. The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$. We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$.

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